THE CLASSIFICATION PROBLEM FOR TORSION-FREE ABELIAN GROUPS ... · free abelian groups A2S(Qn). Here...

JOURNAL OF THEAMERICAN MATHEMATICAL SOCIETYVolume 16, Number 1, Pages 233–258S 0894-0347(02)00409-5Article electronically published on October 8, 2002

THE CLASSIFICATION PROBLEM FOR TORSION-FREEABELIAN GROUPS OF FINITE RANK

SIMON THOMAS

1. Introduction

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-freeabelian group and showed that this notion provided a complete invariant for theclassification problem for torsion-free abelian groups of rank 1. Since then, despitethe efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactorysystem of complete invariants has been found for the torsion-free abelian groupsof finite rank n ≥ 2. So it is natural to ask whether the classification problem isgenuinely more difficult for the groups of rank n ≥ 2. Of course, if we wish toshow that the classification problem for the groups of rank n ≥ 2 is intractible, it isnot enough merely to prove that there are 2ω such groups up to isomorphism. Forthere are 2ω pairwise nonisomorphic groups of rank 1, and we have already pointedout that Baer has given a satisfactory classification for this class of groups. In thispaper, following Friedman-Stanley [11] and Hjorth-Kechris [15], we shall use themore sensitive notions of descriptive set theory to measure the complexity of theclassification problem for the groups of rank n ≥ 2.

Recall that, up to isomorphism, the torsion-free abelian groups of rank n areexactly the additive subgroups of the n-dimensional vector space Qn which containn linearly independent elements. Thus the collection of torsion-free abelian groupsof rank 1 ≤ r ≤ n can be naturally identified with the set S(Qn) of all nontrivialadditive subgroups of Qn. Notice that S(Qn) is a Borel subset of the Polish spaceP(Qn) of all subsets of Qn, and hence S(Qn) can be regarded as a standard Borelspace; i.e. a Polish space equipped with its associated σ-algebra of Borel subsets.(Here we are identifying P(Qn) with the space 2Q

n

of all functions h : Qn → 0, 1equipped with the product topology.) Furthermore, the natural action of GLn(Q)on the vector space Qn induces a corresponding Borel action on S(Qn); and itis easily checked that if A, B ∈ S(Qn), then A ∼= B iff there exists an elementϕ ∈ GLn(Q) such that ϕ(A) = B. It follows that the isomorphism relation onS(Qn) is a countable Borel equivalence relation. (If X is a standard Borel space,then a Borel equivalence relation on X is an equivalence relation E ⊆ X2 which isa Borel subset of X2. The Borel equivalence relation E is said to be countable iffevery E-equivalence class is countable.)

Received by the editors March 1, 2001 and, in revised form, September 25, 2002.2000 Mathematics Subject Classification. Primary 03E15, 20K15, 37A20.Key words and phrases. Borel equivalence relation, torsion-free abelian group, superrigidity.Research partially supported by NSF Grants.

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234 SIMON THOMAS

Notation 1.1. Throughout this paper, the isomorphism relation on S(Qn) will bedenoted by ∼=n.

The notion of Borel reducibility will enable us to compare the complexity of theisomorphism relations for n ≥ 1. If E, F are Borel equivalence relations on thestandard Borel spaces X , Y , respectively, then we say that E is Borel reducible toF and write E ≤B F if there exists a Borel map f : X → Y such that xEy ifff(x)Ff(y). We say that E and F are Borel bireducible and write E ∼B F if bothE ≤B F and F ≤B E. Finally we write E <B F if both E ≤B F and F B E.Most of the Borel equivalence relations that we shall consider in this paper arisefrom group actions as follows. Let G be a lcsc group; i.e. a locally compact sec-ond countable group. Then a standard Borel G-space is a standard Borel space Xequipped with a Borel action (g, x) 7→ g.x of G on X . The corresponding G-orbitequivalence relation on X , which we shall denote by EXG , is a Borel equivalencerelation. In fact, by Kechris [18], EXG is Borel bireducible with a countable Borelequivalence relation. Conversely, by Feldman-Moore [10], if E is an arbitrary count-able Borel equivalence relation on the standard Borel space X , then there exists acountable group G and a Borel action of G on X such that E = EXG .

It is clear that (∼=n) ≤B (∼=n+1) for all n ≥ 1; and our earlier question onthe difficulty of the classification problem for the torsion-free abelian groups ofrank n ≥ 2 can be rephrased as the question of whether (∼=1) <B (∼=n) when n ≥ 2.Before we discuss Hjorth’s solution [14] of this problem and state the main theoremsof this paper, it will be helpful to first give a brief account of some of the generaltheory of countable Borel equivalence relations. (A detailed development of thetheory can be found in Jackson-Kechris-Louveau [16].)

The least complex countable Borel equivalence relations are those which aresmooth; i.e. those countable Borel equivalence relationsE on a standard Borel spaceX for which there exists a Borel function f : X → Y into a standard Borel spaceY such that xEy iff f(x) = f(y). Next in complexity come those countable Borelequivalence relations E such that E is Borel bireducible with the Vitali equivalencerelation E0 defined on 2N by xE0y iff x(n) = y(n) for almost all n. More precisely,by Harrington-Kechris-Louveau [13], if E is a countable Borel equivalence relation,then E is nonsmooth iff E0 ≤B E. Furthermore, by Dougherty-Jackson-Kechris[9], if E is a countable Borel equivalence relation on a standard Borel space X , thenthe following three properties are equivalent:

(1) E ≤B E0.(2) E is hyperfinite; i.e. there exists an increasing sequence

F0 ⊆ F1 ⊆ · · · ⊆ Fn ⊆ · · ·of finite Borel equivalence relations on X such that E =

⋃n∈N Fn. (Here

an equivalence relation F is said to be finite iff every F -equivalence classis finite.)

(3) There exists a Borel action of Z on X such that E = EXZ .It turns out that there is also a most complex countable Borel equivalence rela-

tion E∞, which is universal in the sense that F ≤B E∞ for every countable Borelequivalence relation F , and that E0 <B E∞. (Clearly this universality propertyuniquely determines E∞ up to Borel bireducibility.) E∞ has a number of nat-ural realisations in many areas of mathematics, including algebra, topology andrecursion theory. (See Jackson-Kechris-Louveau [16].) For example, E∞ is Borel


TORSION-FREE ABELIAN GROUPS OF FINITE RANK 235

bireducible to both the isomorphism relation for finitely generated groups [31] andthe isomorphism relation for fields of finite transcendence degree [32].

For many years, it was an open problem whether, up to Borel bireducibility,there existed infinitely many countable Borel equivalence relations E such thatE0 <B E <B E∞. This problem was finally resolved by Adams-Kechris [3], whoused Zimmer’s superrigidity theory [34] to show that there are actually 2ω suchrelations E up to Borel bireducibility. Of these, the only ones which have beenextensively studied are the treeable relations, which were originally introduced byAdams [1]. The countable Borel equivalence relation E on the standard Borel spaceX is said to be treeable iff there exists a Borel relation R ⊆ X ×X such that:

(a) 〈X ;R〉 is an acyclic graph; and(b) the connected components of 〈X ;R〉 are precisely the E-equivalence classes.

For example, whenever a countable free group F has a free Borel action on astandard Borel space X , then the corresponding orbit equivalence relation EXFis treeable. (Here F is said to act freely on X iff g.x 6= x for all 1 6= g ∈ Fand x ∈ X .) The class of treeable relations is not nearly so well understood asthat of the hyperfinite relations. It is known that there exists a universal treeablecountable Borel equivalence relation ET∞; but it remains open whether there existsa (necessarily treeable) countable Borel equivalence relation E such that

E0 <B E <B ET∞,

or even whether ET∞ ≤B E for every non-hyperfinite countable Borel equiva-lence relation E. (A fuller account of the notion of treeability can be found inHjorth-Kechris [15] and Jackson-Kechris-Louveau [16].) We shall return to a fur-ther consideration of these questions at the end of Section 5.

Returning to our discussion of the complexity of the isomorphism relation ∼=n onS(Qn), it is easily checked that Baer’s classification of the rank 1 groups implies that(∼=1) ∼B E0. (For example, see [31] or [21].) In [15], Hjorth-Kechris conjecturedthat (∼=n) ∼B E∞ for all n ≥ 2; in other words, the classification problem forthe torsion-free abelian groups of rank 2 is already as complex as that for arbitraryfinitely generated groups. In [14], Hjorth provided some evidence for this conjectureby proving that E0 <B (∼=n) for all n ≥ 2. (For n ≥ 3, Hjorth actually provedthe stronger result that ∼=n is not treeable. More recently, Kechris [21] has shownthat ∼=2 is also not treeable.) Later Adams-Kechris [3] used Zimmer’s superrigiditytheorem for cocycles [34, Theorem 5.2.5] to prove that

(∼=∗1) <B (∼=∗2) <B · · · <B (∼=∗n) <B · · ·

where (∼=∗n) is the restriction of the isomorphism relation to the class of rigid torsion-free abelian groups A ∈ S(Qn). Here an abelian groupA is said to be rigid if its onlyautomorphisms are the obvious ones: a 7→ a and a 7→ −a. In particular, none of therelations ∼=∗n is a universal countable Borel equivalence relation. It was not clearwhether the Adams-Kechris result should be regarded as evidence for or against theHjorth-Kechris conjecture, since very little was known concerning the relationshipbetween ∼=∗n and ∼=m for n, m ≥ 1. Of course, it is clear that (∼=∗n) ≤B (∼=n) for alln ≥ 1; and it is easily seen that (∼=∗1) ∼B E0 and so (∼=∗1) ∼B (∼=1). But, apart fromthese easy observations, essentially nothing else was known. Recently, Thomas [29]disproved the Hjorth-Kechris conjecture by showing that (∼=∗3) B (∼=2). However,Thomas [29] left open the possibility that there might exist an integer n > 2 such


236 SIMON THOMAS

that (∼=n) ∼B E∞. The main result in this paper shows that none of the relations∼=n is universal.

Theorem 1.2. (∼=∗n+1) B (∼=n) for all n ≥ 1.

As an immediate consequence, we also see that the classification problem forS(Qn+1) is strictly more complex than that for S(Qn).

Theorem 1.3. (∼=n) <B (∼=n+1) for all n ≥ 1.

Theorem 1.2 is an easy consequence of Theorem 1.5. But before we can stateTheorem 1.5, we need to recall some notions from ergodic theory and group theory.Let G be a lcsc group and let X be a standard Borel G-space. Throughout thispaper, a probability measure onX will always mean a Borel probability measure; i.e.a measure which is defined on the collection of Borel subsets of X . The probabilitymeasure µ on X is said to be nonatomic if µ(x) = 0 for all x ∈ X ; and µ is saidto be G-invariant iff µ(g(A)) = µ(A) for every g ∈ G and Borel subset A ⊆ X . TheG-invariant probability measure µ is ergodic iff for every G-invariant Borel subsetA ⊆ X , either µ(A) = 0 or µ(A) = 1. It is well known that the following twoproperties are equivalent:

(i) µ is ergodic.(ii) If Y is a standard Borel space and f : X → Y is a G-invariant Borel

function, then there exists a G-invariant Borel subset M ⊆ X with µ(M) =1 such that f M is a constant function.

For later use, we shall now also recall the notions of a Kazhdan group and anamenable group. Let G be a lcsc group and let π : G→ U(H) be a unitary repre-sentation of G on the separable Hilbert space H. Then π almost admits invariantvectors if for every ε > 0 and every compact subset K ⊆ G, there exists a unitvector v ∈ H such that ||π(g).v − v|| < ε for all g ∈ K. We say that G is a Kazh-dan group if for every unitary representation π of G, if π almost admits invariantvectors, then π has a non-zero invariant vector. If G is a connected semisimpleR-group, each of whose almost R-simple factors has R-rank at least 2, and Γ is alattice in G, then Γ is a Kazhdan group. (For example, see Margulis [26] or Zimmer[34].) In particular, SLm(Z) is a Kazhdan group for each m ≥ 3.

A countable (discrete) group G is amenable if there exists a finitely additiveG-invariant probability measure ν : P(G) → [0, 1] defined on every subset of G.During the proof of Theorem 2.3, we shall make use of the fact that if the countableG is soluble-by-finite, then G is amenable. (For example, see Wagon [33, Theorem10.4].)

In the first three sections of this paper, we shall only discuss countable groupsequipped with the discrete topology. In the later sections, we shall also need toconsider various linear algebraic K-groups G(K) 6 GLn(K), where K is a localfield. Throughout this paper, a field K is said to be local if K is a non-discretelocally compact field of characteristic 0. Each local field K is isomorphic to either R,C or a finite extension of the field Qp of p-adic numbers for some prime p. If K is alocal field and G(K) 6 GLn(K) is an algebraic K-group, then G(K) is a lcsc groupwith respect to the Hausdorff topology; i.e. the topology obtained by restricting thenatural topology on Kn2

to G(K). Any topological notions concerning the groupG(K) will always refer to the Hausdorff topology.



Notation 1.4. Throughout this paper, R(Qn) will denote the Borel set consistingof the groups A ∈ S(Qn) of rank exactly n.

Theorem 1.5. Let m ≥ 3 and let X be a standard Borel SLm(Z)-space withan invariant ergodic probability measure µ. Suppose that 1 ≤ n < m and thatf : X → R(Qn) is a Borel function such that xEXSLm(Z)y implies f(x) ∼=n f(y).Then there exists an SLm(Z)-invariant Borel subset M with µ(M) = 1 such that fmaps M into a single ∼=n-class.

We shall prove Theorem 1.5 in Section 3.

Proof of Theorem 1.2. The case when n = 1 was dealt with in Hjorth [14]. Sosuppose that n ≥ 2. Let R(Qn+1,Zn+1) be the Borel set consisting of those A ∈R(Qn+1) such that Zn+1 6 A. Then R(Qn+1,Zn+1) is invariant under the actionof the subgroup SLn+1(Z) of GLn+1(Q); and Hjorth [14] has shown that thereexists an SLn+1(Z)-invariant Borel subset X of R(Qn+1,Zn+1) with the followingproperties:

(i) Each A ∈ X is rigid.(ii) There exists an SLn+1(Z)-invariant nonatomic probability measure µ on

X .Furthermore, Adams-Kechris [3, Section 6] have shown that we can also supposethat:

(iii) µ is ergodic.Suppose that (∼=∗n+1) ≤B (∼=n). Then there exists a Borel function f : X → S(Qn)such thatA ∼=∗n+1 B iff f(A) ∼=n f(B). By the ergodicity of µ, there exists an integer1 ≤ ` ≤ n and an SLn+1(Z)-invariant Borel subset X0 ⊆ X with µ(X0) = 1 suchthat f(A) has rank ` for each A ∈ X0. Let S`(Qn) be the Borel subset consistingof those A ∈ S(Qn) such that A has rank exactly `. Let g : S`(Qn) → R(Q`) bea Borel function such that C ∼=n D iff g(C) ∼=` g(D) and let h = (g f) X0.Then h : X0 → R(Q`) is a Borel function such that A ∼=∗n+1 B iff h(A) ∼=` h(B).Clearly if A is EX0

SLn+1(Z)-equivalent to B, then A ∼=∗n+1 B and so h(A) ∼=` h(B).Hence by Theorem 1.5, there exists an SLn+1(Z)-invariant Borel subset M ⊆ X0

with µ(M) = 1 such that h maps M into a single ∼=`-class C. But clearly h−1(C)consists of only countably many EX0

SLn+1(Z)-classes, which contradicts the fact thatµ is nonatomic. Hence (∼=∗n+1) B (∼=n).

The above argument also shows that the analogue of Theorem 1.3 holds for theisomorphism relations ∼=n R(Qn) on the groups of rank exactly n. (To see that(∼=n R(Qn)) ≤B (∼=n+1 R(Qn+1)), notice that we can define a Borel reductionf : R(Qn)→ R(Qn+1) by f(A) = A⊕Q.)

The analogue of Theorem 1.3 also holds for the class of p-local torsion-free abeliangroups, which is defined as follows. Throughout this paper, P will denote the set ofprimes. If p ∈ P, then a group A ∈ S(Qn) is said to be p-local iff A = qA for everyprime q 6= p; i.e. A is a Z(p)-module, where

Z(p) = a/b ∈ Q | b is relatively prime to p.

Let S(p)(Qn) and R(p)(Qn) be the Borel sets consisting of the p-local groups A suchthat A ∈ S(Qn), A ∈ R(Qn), respectively; and let ∼=(p)

n be the restriction of theisomorphism relation to S(p)(Qn).


238 SIMON THOMAS

Theorem 1.6. Let p ∈ P be a prime. Then (∼=(p)n ) <B (∼=(p)

n+1) for all n ≥ 1.

Proof. It is easily checked that |S(p)(Q)| = ω and that |S(p)(Q2)| = 2ω; andclearly this implies that (∼=(p)

1 ) <B (∼=(p)2 ). So we can suppose that n ≥ 2. Let

R(p)(Qn+1,Zn+1(p) ) be the Borel set consisting of those A ∈ R(p)(Qn+1) such that

Zn+1(p) 6 A. Then R(p)(Qn+1,Zn+1

(p) ) is invariant under the action of the subgroupSLn+1(Z) of GLn+1(Q); and Hjorth [14] has shown that there exists an SLn+1(Z)-invariant Borel subset X of R(p)(Qn+1,Zn+1

(p) ) with the following properties:

(i) There exists an SLn+1(Z)-invariant nonatomic probability measure µ onX .

(ii) There exists an infinite subgroup L 6 SLn+1(Z) which acts freely on X .

Arguing as in Adams-Kechris [3, Section 6], we can also suppose that:

(iii) µ is ergodic.

Now arguing as in the proof of Theorem 1.2, we see that (∼=(p)n+1) B (∼=n). Hence

(∼=(p)n ) <B (∼=(p)

n+1).

(Once again, it is easily seen that the analogue of Theorem 1.6 also holds for theisomorphism relations on p-local groups of rank exactly n.) Of course, the aboveproof is not completely satisfactory, since it gives no information concerning thecomplexity of the relation ∼=(p)

2 . In particular, it leaves open the possibility that∼=(p)

2 is smooth. This situation will be remedied in Section 5, where we shall showthat ∼=(p)

2 is not treeable.In an earlier version of this paper, I conjectured that if p 6= q are distinct primes

and n ≥ 2, then the isomorphism relations ∼=(p)n and ∼=(q)

n are incomparable withrespect to Borel reducibility. In Thomas [30], this was proved in the case whenn ≥ 3. However, the case when n = 2 still remains open.

This paper is organised as follows. In Section 2, we shall discuss the notionof a cocycle of a group action and state the two cocycle reduction results whichare needed in the proof of Theorem 1.5. In Sections 3 and 4, we shall consider thequasi-equality and quasi-isomorphism relations onR(Qn). These relations were firstintroduced by Jonsson [17], who showed that much of the pathological behaviour ofthe class of finite rank torsion-free abelian groups disappears when the isomorphismrelation is replaced by the coarser quasi-isomorphism relation. In Section 3, we shallinitially prove the analogue of Theorem 1.5 for the quasi-isomorphism relation,modulo the result that the quasi-equality relation is hyperfinite, which will beproved in Section 4. Theorem 1.5 will then follow easily. In Section 5, we shallstudy the class of p-local torsion-free abelian groups of finite rank. In particular,we shall show that the isomorphism relation for the class of p-local groups of rank2 is not treeable. Finally in Section 6, we shall prove our main cocycle reductionresult.

Throughout this paper, we shall identify linear transformations π : Qn → Qnwith the corresponding matrices Mπ ∈Matn(Q) with respect to the standard basise1, . . . , en. If π ∈Matn(Q), then πt denotes the transpose of π. If J is a ring, thenJ∗ denotes the group of multiplicative units of J . If K is a field, then K denotesthe algebraic closure of K.



2. Cocycles

In this section, we shall discuss the notion of a cocycle of a group action andstate the two cocycle reduction results which are needed in the proof of Theorem1.5. (Clear accounts of the theory of cocycles can be found in Zimmer [34] andAdams-Kechris [3]. In particular, Adams-Kechris [3, Section 2] provides a conve-nient introduction to the basic techniques and results in this area, written for thenon-expert in the ergodic theory of groups.) Let G be a lcsc group and let X be astandard Borel G-space with an invariant probability measure µ.

Definition 2.1. If H is a lcsc group, then a Borel function α : G × X → H iscalled a cocycle if for all g, h ∈ G,

α(hg, x) = α(h, g.x)α(g, x) µ-a.e.(x).

If the above equation holds for all g, h ∈ G and all x ∈ X , then α is said to be astrict cocycle. By Zimmer [34, Theorem B.9], if α : G×X → H is a Borel cocycle,then there exists a strict cocycle β : G×X → H such that for all g ∈ G,

β(g, x) = α(g, x) µ-a.e.(x).

The standard example of a cocycle arises in the following fashion. Suppose that Yis a standard Borel H-space and that H acts freely on Y . If f : X → Y is a Borelfunction such that xEXG y implies f(x)EYHf(y), then we can define a strict Borelcocycle α : G×X → H by letting α(g, x) be the unique element of H such that

α(g, x).f(x) = f(g.x).

Suppose now that B : X → H is a Borel function and that f ′ : X → Y isdefined by f ′(x) = B(x).f(x). Then xEXG y also implies that f ′(x)EYHf

′(y); andthe corresponding cocycle α′ : G×X → H satisfies

α′(g, x) = B(g.x)α(g, x)B(x)−1

for all g ∈ G and x ∈ X . This observation motivates the following definition.

Definition 2.2. Let H be a lcsc group. Then the cocycles α, β : G×X → H areequivalent , written α ∼ β, iff there exists a Borel function B : X → H such thatfor all g ∈ G,

β(g, x) = B(g.x)α(g, x)B(x)−1 µ-a.e.(x).

The proof of Theorem 1.5 is based upon a number of cocycle reduction results;i.e. theorems which say that under suitable hypotheses on G and H , every cocycleα : Γ × X → H is equivalent to a cocycle β such that β(Γ × X) is contained ina “small” subgroup of H . Before we state our main cocycle reduction theorem,we need to review some notions from the theory of algebraic groups. Throughoutthis paper, Ω will denote a fixed algebraically closed field of characteristic 0 whichcontains R and all of the p-adic fields Qp. By an algebraic group, we shall alwaysmean a subgroup G of some general linear group GLn(Ω) which is Zariski closed inGLn(Ω). If G can be defined using polynomials with coefficients in the subfield kof Ω, then we say that G is an algebraic k-group. For any subring R ⊆ Ω, we define

GLn(R) = (rij) ∈ GLn(Ω) | rij ∈ R and det(rij)−1 ∈ R;and if G 6 GLn(Ω) is an algebraic group, then we define

G(R) = G ∩GLn(R).


240 SIMON THOMAS

In particular, if R = k is a subfield of Ω and G is an algebraic k-group, then G(k)is the group of all matrices in G with entries in k.

The following cocycle reduction theorem is a straightforward consequence ofZimmer’s superrigidity theorem [34, Theorem 5.2.5] and the ideas of Adams-Kechris[3].

Theorem 2.3. Let m ≥ 3 and let X be a standard Borel SLm(Z)-space withan invariant ergodic probability measure. Suppose that G is an algebraic Q-groupsuch that dimG < m2 − 1 and that H 6 G(Q). Then for every Borel cocycleα : SLm(Z)×X → H, there exists an equivalent cocycle γ such that γ(SLm(Z)×X)is contained in a finite subgroup of H.

We shall prove Theorem 2.3 in Section 6. In the proof of Theorem 1.5, we shallalso make use of the following cocycle reduction result, which is a special case ofHjorth-Kechris [15, Theorem 10.5]. (It should be pointed out that this particularspecial case is a consequence of Zimmer [34, Theorem 9.1.1].)

Theorem 2.4. Let m ≥ 3 and let X be a standard Borel SLm(Z)-space with aninvariant ergodic probability measure µ. Suppose that Y is a standard Borel spaceand that F is a hyperfinite equivalence relation on Y . If f : X → Y is a Borelfunction such that xEXSLm(Z)y implies f(x)Ff(y), then there exists an SLm(Z)-invariant Borel subset M with µ(M) = 1 such that f maps M into a single F -class.

3. The quasi-isomorphism relation

In this section, we shall use our cocycle reduction theorems to prove Theorem 1.5.We shall begin by saying a few words about the strategy of the proof. So supposethat m ≥ 3 and that X is a standard Borel SLm(Z)-space with an invariant ergodicprobability measure µ. Suppose that 1 ≤ n < m and that f : X → R(Qn) is aBorel function such that xEXSLm(Z)y implies f(x) ∼=n f(y). Because GLn(Q) doesnot act freely on R(Qn), we are initially unable to define a corresponding cocycleα : SLm(Z) ×X → GLn(Q).

In Thomas [29], we were able to get around this difficulty, in the case whenn = 2, by reducing to the situation where there exists a Borel subset X0 ⊆ X withµ(X0) = 1 such that Aut(f(x)) is a fixed subgroup L of GL2(Q) for all x ∈ X0.This meant that ∼=2 f(X0) was induced by a free action of the quotient group

H = NGL2(Q)(L)/L;

and so we could define a corresponding cocycle α : SLm(Z) ×X → H . However,this would not have been useful unless there existed a suitable cocycle reductionresult for cocycles taking values in H ; and for such a result to exist, it was necessarythat H should be a “reasonably classical” group. Fortunately, using Krol’s analysis[22] of the automorphism groups of the torsion-free abelian groups of rank 2, wewere able to explicitly enumerate the possibilities for L; and it turned out that wecould make a further reduction to the situation when H was a slight variant ofPGL2(Q).

Now suppose that n > 2. Then it seems likely that, using Arnold [4], we canonce again reduce to the situation where there exists a Borel subset X0 ⊆ X withµ(X0) = 1 such that Aut(f(x)) is a fixed subgroup L of GLn(Q) for all x ∈X0. Unfortunately, as n gets larger, the possibilities for L become more and morecomplex. In fact, by Corner’s Theorem [7], L can be isomorphic to the group of units



of an arbitrary reduced subring of Matd(Q), where d = bn/2c; and it is far from clearwhether H = NGLn(Q)(L)/L will always be a “reasonably classical” group. In orderto avoid these algebraic complications, we shall initially replace the isomorphismrelation on R(Qn) by the coarser relation of quasi-isomorphism. This relation wasfirst introduced in Jonsson [17], where it was shown that the class of torsion-freeabelian groups of finite rank has a better decomposition theory with respect toquasi-isomorphism than with respect to isomorphism. This decomposition theorywill not concern us in this paper. Rather we shall exploit the fact that muchof the number-theoretical complexity of a finite rank torsion-free abelian group islost when we work with respect to quasi-isomorphism; and this turns out to beenough to ensure that the cocycles that arise in our analysis always take valuesin a “reasonably classical” group. As a bonus, we obtain that the problem ofclassifying the torsion-free abelian groups of finite rank up to quasi-isomorphismis also intractible. (However, we should point out that the shift from isomorphismto quasi-isomorphism comes at a cost. In Thomas [29], the proof yields an explicitdecomposition of ∼=2 as a direct sum of amenable relations and orbit relationsinduced by free actions of homomorphic images of GL2(Q). It does not seempossible to extract an analogous decomposition of ∼=n from the current proof.)

Definition 3.1. Suppose that A, B ∈ R(Qn). Then A is said to be quasi-containedin B, written A ≺n B, if there exists an integer m > 0 such that mA 6 B. IfA ≺n B and B ≺n A, then A and B are said to be quasi-equal and we writeA ≈n B.

Recall that if A ∈ R(Qn) and m > 0, then [A : mA] <∞. (For example, see [12,Exercise 92.5].) It follows that if A, B ∈ R(Qn), then A ≈n B iff A ∩ B has finiteindex in both A and B.

Lemma 3.2. ≈n is a countable Borel equivalence relation on R(Qn).

Proof. It is clear that ≈n is a Borel equivalence relation. Thus it is enough to showthat if A ∈ R(Qn), then there only exist countably many B ∈ R(Qn) such thatA ≈n B. To see this, suppose that A ≈n B and let r, s > 0 be integers such thatrA 6 B and sB 6 A. Then rA 6 B 6 (1/s)A and

[(1/s)A : rA] = [A : rsA] <∞.

It follows that there are only countably many possibilities for B.

Definition 3.3. Suppose that A, B ∈ R(Qn). Then A and B are said to be quasi-isomorphic, written A ∼n B, if there exists ϕ ∈ GLn(Q) such that ϕ(A) ≈n B.

Using Lemma 3.2, it follows that each ∼n-class consists of only countably many∼=n-classes. In particular, ∼n is also a countable Borel equivalence relation onR(Qn). Most of our effort in this section will be devoted to proving the followinganalogue of Theorem 1.5 for the quasi-isomorphism relation.

Theorem 3.4. Let m ≥ 3 and let X be a standard Borel SLm(Z)-space withan invariant ergodic probability measure µ. Suppose that 1 ≤ n < m and thatf : X → R(Qn) is a Borel function such that xEXSLm(Z)y implies f(x) ∼n f(y).Then there exists an SLm(Z)-invariant Borel subset M with µ(M) = 1 such that fmaps M into a single ∼n-class.


242 SIMON THOMAS

Arguing as in the proof of Theorem 1.2, we now easily obtain the following result.

Theorem 3.5. (∼n) <B (∼n+1) for all n ≥ 1.

(Of course, since ∼n is defined to be the quasi-isomorphism relation on the spaceR(Qn) of groups of rank exactly n, it is necessary to explain why (∼n) ≤B (∼n+1).To see this, let f : R(Qn)→ R(Qn+1) be the Borel map defined by f(A) = A⊕Q.Using Fuchs [12, Theorem 92.5], it follows easily that f is a Borel reduction from∼n to ∼n+1.) Before we begin the proof of Theorem 3.4, we shall show how toderive Theorem 1.5.

Proof of Theorem 1.5. Suppose that f : X → R(Qn) is a Borel function such thatxEXSLm(Z)y implies f(x) ∼=n f(y). Then obviously xEXSLm(Z)y implies f(x) ∼n f(y);and so by Theorem 3.4, there exists an SLm(Z)-invariant Borel subset Y withµ(Y ) = 1 such that f maps Y into a single ∼n-class C. Since C is countable, thereexists a Borel subset Z ⊆ Y with µ(Z) > 0 and a fixed group A ∈ C such thatf(x) = A for all x ∈ Z. Since µ is ergodic, the SLm(Z)-invariant Borel subsetM = SLm(Z).Z satisfies µ(M) = 1; and clearly f maps M into the ∼=n-classcontaining A.

For each A ∈ R(Qn), let [A] be the ≈n-class containing A. During the proof ofTheorem 3.4, we shall consider the action of GLn(Q) on the set of ≈n-classes. Inorder to compute the setwise stabiliser in GLn(Q) of a ≈n-class [A] , it is necessaryto introduce the notions of a quasi-endomorphism and a quasi-automorphism. IfA ∈ R(Qn), then a linear transformation ϕ ∈ Matn(Q) is said to be a quasi-endomorphism of A iff ϕ(A) ≺n A. Equivalently, ϕ is a quasi-endomorphism of Aiff there exists an integer m > 0 such that mϕ ∈ End(A). It is easily checked thatthe collection QE(A) of quasi-endomorphisms of A is a Q-subalgebra of Matn(Q)and that if A ≈n B, then QE(A) = QE(B). A linear transformation ϕ ∈ Matn(Q)is said to be a quasi-automorphism of A iff ϕ is a unit of the Q-algebra QE(A).The group of quasi-automorphisms of A is denoted by QAut(A). By Exercise 6.1[4], if ψ ∈ End(A), then ψ ∈ QAut(A) iff ψ is a monomorphism.

Lemma 3.6. If A ∈ R(Qn), then QAut(A) is the setwise stabiliser of [A] inGLn(Q).

Proof. First suppose that ϕ ∈ QAut(A). Then there exists an integer m > 0such that ψ = mϕ ∈ End(A). Clearly ψ is also a unit of QE(A) and so ψ is amonomorphism. Hence by [12, Exercise 92.5], ψ(A) has finite index in A and soψ(A) ≈n A. Since ψ(A) = mϕ(A), it follows that ψ(A) ≈n ϕ(A). Thus ϕ(A) ≈n Aand so ϕ stabilises [A].

Conversely suppose that ϕ ∈ GLn(Q) stabilises [A]. Then ϕ(A) ≈n A and sothere exists an integer m > 0 such that mϕ(A) 6 A. Since mϕ ∈ End(A) is amonomorphism, it follows that mϕ ∈ QAut(A) and so ϕ ∈ QAut(A).

Now we are ready to begin the proof of Theorem 3.4. So let m ≥ 3 and let Xbe a standard Borel SLm(Z)-space with an invariant ergodic probability measureµ. Suppose that 1 ≤ n < m and that f : X → R(Qn) is a Borel function such thatxEXSLm(Z)y implies f(x) ∼n f(y). Let E = EXSLm(Z) and for each x ∈ X , let Ax =f(x) ∈ R(Qn). First notice that there are only countably many possibilities for theQ-algebra QE(Ax). Hence there exists a Borel subset X1 ⊆ X with µ(X1) > 0 anda fixed Q-subalgebra S of Matn(Q) such that QE(Ax) = S for all x ∈ X1. By the



ergodicity of µ, we have that µ(SLm(Z).X1) = 1. In order to simplify notation, weshall assume that SLm(Z).X1 = X . After slightly adjusting f if necessary, we cansuppose that QE(Ax) = S for all x ∈ X . (More precisely, let c : X → X be a Borelfunction such that c(x)Ex and c(x) ∈ X1 for all x ∈ X . Then we can replace fwith f ′ = f c.) In particular, we have that QAut(Ax) = S∗, the group of unitsof S, for each x ∈ X . Now suppose that x, y ∈ X and that xEy. Then Ax ∼n Ayand so there exists ϕ ∈ GLn(Q) such that ϕ(Ax) ≈n Ay. Notice that

ϕSϕ−1 = ϕQE(Ax)ϕ−1 = QE(ϕ(Ax)) = QE(Ay) = S

and so ϕ ∈ N = NGLn(Q)(S). Clearly we also have that ϕ([Ax]) = [Ay]; and byLemma 3.6, for each x ∈ X , the stabiliser of [Ax] in GLn(Q) is QAut(Ax) = S∗.Let H = N/S∗ and for each ϕ ∈ N , let ϕ = ϕS∗. Then we can define a Borelcocycle α : SLm(Z) ×X → H by

α(g, x) = the unique element ϕ ∈ H such that ϕ([Ax]) = [Ag.x].

Lemma 3.7. There exists an algebraic Q-group G with dimG < m2 − 1 such thatH 6 G(Q).

Proof. Recall that throughout this paper, Ω denotes a fixed algebraically closedfield containing R and all of the p-adic fields Qp. Let

Λ = Ω⊗ S ⊆Matn(Ω)

be the associated Ω-algebra. Then Λ is an affine Q-variety; and the Cayley-Hamilton Theorem implies that the group of units of Λ is given by

Λ∗ = ϕ ∈ Λ | det(ϕ) 6= 0.

Thus Λ∗ is an algebraic Q-group and Λ∗(Q) = S∗. Furthermore, by [6, Proposition1.7], Γ = NGLn(Ω)(Λ) is also an algebraic Q-group and clearly Γ(Q) = N . By [6,Theorem 6.8], G = Γ/Λ∗ is an algebraic Q-group and

H = Γ(Q)/Λ∗(Q) 6 G(Q).

Finally note that

dimG ≤ dim Γ ≤ dimGLn(Ω) = n2 < m2 − 1.

By Theorem 2.3, α is equivalent to a cocycle γ such that γ(SLm(Z) × X) iscontained in a finite subgroup K of H . Let B : X → H be a Borel function suchthat:

(*) for all g ∈ SLm(Z), α(g, x) = B(g.x)γ(g, x)B(x)−1 µ-a.e.(x).

It is easily checked that if x satisfies (*) and xEy, then y also satisfies (*). Tosimplify notation, we shall assume that (*) holds for all x ∈ X . Now there existsa Borel subset X1 ⊆ X with µ(X1) > 0 and a fixed element ψ ∈ H such thatB(x) = ψ for all x ∈ X1. Since µ is ergodic, µ(SLm(Z).X1) = 1 and so we can alsoassume that X1 intersects every SLm(Z)-orbit on X . Let c : X → X be a Borelfunction such that c(x)Ex and c(x) ∈ X1 for each x ∈ X ; and let x1 = c(x). Thenfor each x ∈ X ,

Ψ(x) = α(g, x1) | g.x1 ∈ X1 ⊆ ψKψ−1


244 SIMON THOMAS

is a nonempty finite subset of H . Hence for each x ∈ X ,

Φ(x) = ϕ([Ax1 ]) | ϕ = α(g, x1) for some g.x1 ∈ X1= [Ag.x1 ] | g.x1 ∈ X1= [Ay] | yEx and y ∈ X1

is a nonempty finite set of ≈n-classes; and clearly if xEy, then Φ(x) = Φ(y). Bythe ergodicity of µ, we can suppose that there exists an integer 1 ≤ k ≤ |K| suchthat |Φ(x)| = k for all x ∈ X . Now let x 7→ (x1, . . . , xk) be a Borel function fromX to Xk such that for each x ∈ X ,

(a) xiEx and xi ∈ X1; and(b) Φ(x) = [Ax1 ], . . . , [Axk ].

Finally let f : X → R(Qn)k be the Borel function defined by

f(x) = (Ax1 , . . . , Axk);

and let F be the countable Borel equivalence relation on R(Q)k defined by

(A1, . . . , Ak)F (B1, . . . , Bk) iff [A1], . . . , [Ak] = [B1], . . . , [Bk].

To complete the proof of Theorem 3.4, we now require the following theorem, whichwill be proved in Section 4.

Theorem 3.8. For each n ≥ 1, the relation ≈n is hyperfinite.

Using the fact that ≈n is hyperfinite, it follows easily that F is also hyperfinite.(For example, see [16, Section 1].) Notice that if xEy, then Φ(x) = Φ(y) and sof(x)F f (y). By Theorem 2.4, there exists an SLm(Z)-invariant Borel subset M ⊆ Xwith µ(M) = 1 such that f maps M into a single F -class; and this implies that fmaps M into a single ∼n-class. This completes the proof of Theorem 3.4.

Finally we should point out that both of the following questions remain open.

Question 3.9. Is (∼=n) ≤B (∼n) for n ≥ 2?

Question 3.10. Is (∼n) ≤B (∼=n) for n ≥ 2?

In an earlier version of this paper, I pointed out that a negative answer toQuestion 3.9 would be especially interesting, since it was then unknown whetherthere existed a pair E, F of countable Borel equivalence relations on a standardBorel space X such that E ⊆ F and E B F . Soon afterwards, Adams [2] provedthat there exists a pair E ⊆ F of countable Borel equivalence relations such thatE and F are incomparable with respect to Borel reducibility.

4. The hyperfiniteness of the quasi-equality relation

In this section, we shall prove Theorem 3.8 which says that the quasi-equalityrelation ≈n is hyperfinite for each n ≥ 1. The following lemma, which is due toLady [24], will enable us to restrict our attention to the class of p-local groups.Recall that Z(p) is the ring of rational numbers a/b ∈ Q such that b is relativelyprime to p.

Definition 4.1. If A ∈ R(Qn) and p ∈ P, then the localisation of A at p is definedto be Ap = Z(p) ⊗A.



Lemma 4.2. If A, B ∈ R(Qn), then A ≈n B iff the following two conditions aresatisfied:

(i) Ap ≈n Bp for all primes p ∈ P; and(ii) Ap = Bp for all but finitely many primes p ∈ P.

Definition 4.3. For each prime p ∈ P, we define ≈(p)n to be the restriction of the

quasi-equality relation to the space R(p)(Qn) of p-local groups A ∈ R(Qn).

Most of this section will be devoted to proving the following special case ofTheorem 3.8.

Lemma 4.4. For each prime p ∈ P, the relation ≈(p)n is smooth.

Before proving Lemma 4.4, we shall show how to complete the proof of Theorem3.8.

Proof of Theorem 3.8. Let E0 be the Vitali equivalence relation on 2N; and let E∗0be the corresponding equivalence relation on 2P×N, defined by xE∗0y iff x(p, n) =y(p, n) for all but finitely many pairs (p, n) ∈ P× N. Then clearly E∗0 ∼B E0. Foreach prime p ∈ P, since the relation ≈(p)

n is smooth, there exists an injective Borelmap gp : R(p)(Qn)→ 2N such that A ≈(p)

n B iff gp(A)E0gp(B). Consider the Borelmap f : R(Qn) → 2P×N, defined by f(A)(p, n) = gp(Ap)(n). By Lemma 4.2, if A,B ∈ R(Qn), then A ≈n B iff f(A)E∗0f(B). Hence ≈n is a hyperfinite equivalencerelation.

Thus it only remains to prove Lemma 4.4. For the rest of this section, weshall fix a prime p ∈ P and, to simplify the notation, we shall write ≈ instead of≈(p)n . Throughout this section, we shall regard Qn as an additive subgroup of the

n-dimensional vector space Qnp over the field Qp of p-adic numbers; and we shallextend the relation ≈ to the collection of all additive subgroups of Qnp by settingC ≈ D iff C ∩D has finite index in both C and D. Let Zp be the ring of p-adicintegers. Following the approach of Kurosh [23] and Malcev [25], we shall nowfurther localise each p-local group A ∈ R(p)(Qn) to a corresponding Zp-submoduleA of Qnp . The motivation for this is that while A might have a very complexstructure, A will always decompose into a direct sum of copies of Zp and Qp.

Definition 4.5. For each A ∈ R(p)(Qn), we define A = Zp ⊗A.

We shall regard each A as a subgroup of Qnp in the usual way; i.e. A is thesubgroup consisting of all finite sums

γ1a1 + γ2a2 + · · ·+ γtat,

where γi ∈ Zp and ai ∈ A for 1 ≤ i ≤ t. By [12, Lemma 93.3], there exist integers0 ≤ k, ` ≤ n with k + ` = n and elements vi, wj ∈ A such that

A =k⊕i=1

Qpvi ⊕⊕j=1

Zpwj .

Definition 4.6. For each A ∈ R(p)(Qn), we define VA =⊕k

i=1Qpvi.

The following result clarifies the algebraic content of the quasi-equality relationon R(p)(Qn).


246 SIMON THOMAS

Theorem 4.7. If A, B ∈ R(p)(Qn), then A ≈ B iff VA = VB .

Theorem 4.7 is an immediate consequence of the following two lemmas.

Lemma 4.8. If A, B ∈ R(p)(Qn), then A ≈ B iff A ≈ B.

Proof. First suppose that A ≈ B. By [12, Lemma 93.2], we have that A ∩Qn = A

and B ∩Qn = B. Thus

[A : A ∩B] = [A ∩Qn : (A ∩ B) ∩Qn] ≤ [A : A ∩ B] <∞.Similarly, [B : A ∩B] <∞ and hence A ≈ B.

Conversely suppose that A ≈ B and let C = A∩B. Then C 6 A∩ B and so it isenough to show that C has finite index in both A and B. To see this, let F = A/Cand consider the short exact sequence

0→ C → A→ F → 0.

Then by [12, Theorem 60.2], the sequence

Zp ⊗ C → Zp ⊗A→ Zp ⊗ F → 0

is also exact. Let F =⊕s

r=1 Cr be a decomposition of F into a direct sum of finitecyclic groups Cr of order mr. Then by [12, Section 59],

Zp ⊗ F ∼=s⊕r=1

Zp/mrZp.

Thus Zp⊗F is a finite group and so C has finite index in A. Similarly C has finiteindex in B.

Lemma 4.9. If A, B ∈ R(p)(Qn), then A ≈ B iff VA = VB.

Proof. First suppose that A ≈ B. Then for each v ∈ VA, we have that

[Qpv : B ∩Qpv] = [A ∩Qpv : A ∩ B ∩Qpv] ≤ [A : A ∩ B] <∞.

Suppose that v ∈ VArVB. Then B ∩Qpv is a proper Zp-submodule of Qpv and sothere exists a vector 0 6= u ∈ Qpv such that B ∩Qpv = Zpu. But then

Qpv/(B ∩Qpv) ∼= Qp/Zp ∼= C(p∞),

which is a contradiction. Thus VA 6 VB. Similarly VB 6 VA and so VA = VB.Conversely suppose that VA = VB = V . Then there exists an integer 0 ≤ ` ≤ n

and elements xj ∈ A, yj ∈ B such that

A = V ⊕⊕j=1

Zpxj

and

B = V ⊕⊕j=1

Zpyj .

Let LA =⊕`

j=1 Zpxj and LB =⊕`

j=1 Zpyj . Then we can identify LA, LB withthe corresponding Zp-submodules of the `-dimensionalQp-vector space W = Qnp/V .



Now there exists an integer t ≥ 0 such that ptyj ∈ LA for each 1 ≤ j ≤ `. It followsthat

[LB : LA ∩ LB] ≤ [LB : ptLB] = pt`.

Similarly [LA : LA ∩ LB] <∞ and hence A ≈ B. Now that we have dealt with the purely algebraic aspect of the quasi-equality

relation on R(p)(Qn), we shall next consider its descriptive set-theoretic aspect.Recall that the vector space Qnp is a complete separable metric space with respectto the metric induced by the usual p-adic norm; and that each Qp-vector subspaceV 6 Qnp is a closed subset of Qnp with respect to this metric. The Effros Borel spaceon Qnp is defined to be the set

F (Qnp ) = Z ⊆ Qnp | Z is a closed subset of Qnpequipped with the σ-algebra generated by the sets of the form

Z ∈ F (Qnp ) | Z ∩ U 6= ∅,where U varies over the open subsets of Qnp . By [20, Theorem 12.6], F (Qnp ) is astandard Borel space. Thus to complete the proof of Theorem 3.8, we need onlyshow that the map A 7→ VA is a Borel map fromR(p)(Qn) into F (Qnp ). Furthermore,it is clear that the map s : (Qnp )≤n → F (Qnp ), defined by

s(v1, . . . , vk) = the Qp-subspace spanned by v1, . . . , vk,is Borel. (For example, this follows easily from [20, Exercise 12.14].) Hence we needonly show that there exists a Borel map b : R(p)(Qn) → (Qnp )≤n such that b(A) isa basis of VA. (The details of the following construction of a basis b(A) of VA willbe used in the next section, in which we study the structure of a “random” groupA ∈ R(p)(Qn,Zn(p)).)

Definition 4.10. Let A ∈ R(p)(Qn). Then a sequence (a1, . . . , a`) of nonzeroelements of A is said to be p-independent iff whenever n1, . . . , n` ∈ Z are such that

n1a1 + · · ·+ n`a` ∈ pA,then p divides nj for all 1 ≤ j ≤ `. The sequence (a1, . . . , a`) is said to be a p-basisiff (a1, . . . , a`) is a maximal p-independent sequence.

Fix some A ∈ R(p)(Qn). Then we can clearly choose a p-basis (a1, . . . , a`) of Ain a Borel fashion. Let P = 〈a1, . . . , a`〉 be the subgroup generated by a1, . . . , a`.Then by [12, Section 32], A/P is p-divisible and so A/P is a divisible group. ThusA/P = R ⊕ T , where T is the torsion subgroup and R is the direct sum of k =n − ` copies of Q. Furthermore, by [12, Exercise 93.1], dimVA = k. Next, ina Borel fashion, we can choose a sequence (z1, . . . , zk) of elements of A such that(z1P, . . . , zkP ) is a basis ofR ∼= Qk. Finally we shall use (z1, . . . , zk) and (a1, . . . , a`)to construct a basis (v1, . . . , vk) of VA.

Fix some 1 ≤ i ≤ k. Let t ≥ 1 and suppose inductively that there exist integersc(j)s for 1 ≤ j ≤ ` and 1 ≤ s ≤ t such that the following conditions are satisfied.

(1) 0 ≤ c(j)s < p.(2) There exists dt ∈ A such that

ptdt = zi + n(1)t a1 + · · ·+ n

(`)t a`,

where n(j)t = c

(j)1 + c

(j)2 p+ · · ·+ c

(j)t pt−1.


248 SIMON THOMAS

Since R is divisible, there exists an element dt+1 ∈ A and integers c(1)t+1, . . . , c

(`)t+1

such thatpdt+1 = dt + c

(1)t+1a1 + · · ·+ c

(`)t+1a`;

and after adjusting our choice of dt+1 if necessary, we can suppose that 0 ≤ c(j)t+1 < pfor each 1 ≤ j ≤ `. Clearly

pt+1dt+1 = zi + n(1)t+1a1 + · · ·+ n

(`)t+1a`.

Thus the induction can be completed. For 1 ≤ j ≤ `, let

γj = c(j)1 + c

(j)2 p+ · · ·+ c

(j)t pt−1 + · · · ∈ Zp.

Then the corresponding basis element of VA is

vi = zi + γ1a1 + · · ·+ γ`a` ∈ A.Since z1, . . . , zk are linearly independent over Qp, it follows that v1, . . . , v` are alsolinearly independent over Qp. Thus it is enough to check that vi ∈ ptA for each1 ≤ i ≤ k and t ≥ 1. So fix some 1 ≤ i ≤ k and reconsider the element

vi = zi + γ1a1 + · · ·+ γ`a` ∈ A.If t ≥ 1, then

ptdt = zi + n(1)t a1 + · · ·+ n

(`)t a`

and

vi − ptdt = (∞∑

r=t+1

c(1)r pr−1)a1 + · · ·+ (

∞∑r=t+1

c(`)r pr−1)a`.

Hence ptet = vi, where

et = dt + (∞∑

r=t+1

c(1)r pr−(t+1))a1 + · · ·+ (

∞∑r=t+1

c(`)r pr−(t+1))a`.

This completes the proof of Theorem 3.8.

5. p-local torsion-free abelian groups of finite rank

Fix some n ≥ 2 and letR = R(p)(Qn,Zn(p)) be the standard Borel space consistingof those A ∈ R(Qn) such that Zn(p) 6 A. Then by [14, Lemma 4.11], R ⊆ R(p)(Qn).Furthermore,R is invariant under the action of the subgroupGLn(Z(p)) of GLn(Q);and Hjorth [14] has shown that there exists a GLn(Z(p))-invariant probability mea-sure µ on R. Each A ∈ R has certain obvious automorphisms; namely, for eachs ∈ Z∗(p), we can define a corresponding automorphism of A by a 7→ sa. So if weidentify each s ∈ Z∗(p) with the corresponding diagonal matrix Ds ∈ GLn(Q), wehave that

Z∗(p) 6 Aut(A) 6 GLn(Q).Our main result in this section says that a “random” group A ∈ R has only theseobvious automorphisms.

Theorem 5.1. µ(A ∈ R | Aut(A) = Z∗(p)) = 1.

Using Theorem 5.1 and Kechris’s result [21] that PSL2(Z[1/q]) is antitreeablefor every prime q ∈ P, we can now easily prove that ∼=(p)

2 is not treeable. (Recallthat in [14], Hjorth proved that ∼=(p)

n is not treeable for each n ≥ 3.)



Definition 5.2. A countable group G is said to be antitreeable iff for every Borelaction of G on a standard Borel space X , which is free and admits an invariant Borelprobability measure, the correponding equivalence relation EXG is not treeable.

Corollary 5.3. ∼=(p)2 is not treeable.

Proof. Let q ∈ P be a prime such that q 6= p and let

G = PSL2(Z[1/q]) 6 PGL2(Z(p)).

By Theorem 5.1, there exists a GL2(Z(p))-invariant Borel subset X ⊆ R(p)(Q2,Z2(p))

with µ(X) = 1 such that Aut(A) = Z∗(p) for all A ∈ X . Now the µ-preserving actionof PGL2(Z(p)) on R(p)(Q2,Z2

(p)) induces a corresponding free action of PGL2(Z(p))on X ; and hence there exists a µ-preserving free action of G on X . By [21, Theorem7], G is antitreeable and so EXG is not treeable. Since EXG ⊆ (∼=(p)

2 X), [14, Lemma2.10] implies that ∼=(p)

2 is also not treeable.

Before we begin the proof of Theorem 5.1, we shall review Hjorth’s construction[14] of the measure µ on R. Let S be the standard Borel space consisting of allsubgroups of the quotient groupQn/Zn(p) and let π : Qn → Qn/Zn(p) be the canonicalsurjection. Then we can define a Borel bijection R → S by A 7→ π(A). Let Γ be thedual group of Qn/Zn(p); i.e. the space of all homomorphisms ψ : Qn/Zn(p) → R/Z,equipped with the topology of pointwise convergence and the group operation ofpointwise addition. Since Qn/Zn(p) is discrete, it follows that Γ is a compact group.Let ν be the Haar measure on Γ. Let k : Γ → S be the Borel map assigning thesubgroup

ker(ψ) = h ∈ Qn/Zn(p) | ψ(h) = 0to each ψ ∈ Γ; and let µ = kν be the probability measure defined on S by

µ(B) = ν(ψ ∈ Γ | ker(ψ) ∈ B)for each Borel subset B ⊆ S. Then µ is the corresponding Borel probability measureon R induced by the Borel bijection A 7→ π(A).

During the proof of Theorem 5.1, we shall make use of the fact that Γ is naturallyisomorphic to Znp . To see this, first notice that since Qn/Zn(p) ∼= C(p∞)n, it followsthat each ψ ∈ Γ must take values in Z[1/p]/Z ∼= C(p∞). For each v ∈ Qn anda ∈ Z[1/p], we shall denote the corresponding elements of Qn/Zn(p) and Z[1/p]/Zby [v], [a], respectively. Let e1, . . . , en be the standard basis of Qn. Then we candefine an isomorphism ψ 7→ ψ of Γ onto Znp by

ψ = (β1, . . . , βn),

where βi = b(i)1 + b

(i)2 p+ · · ·+ b

(i)t pt−1 + · · · is the p-adic integer such that

ψ([ei/pt]) =

[b(i)1 + b

(i)2 p+ · · ·+ b

(i)t pt−1

pt

]for each t ≥ 1. We shall also denote the Haar measure on Znp by ν.

For each A ∈ R, let VA 6 A = Zp ⊗ A be the Qp-vector space defined inDefinition 4.6. Then 0 ≤ dimVA ≤ n. We shall begin our analysis by determiningthe value of dimVA for a “random” subgroup A ∈ R.


250 SIMON THOMAS

Definition 5.4. Let R1 ⊂ R be the Borel subset consisting of those A ∈ R whichsatisfy the following conditions:

(i) For all 0 6= a ∈ A, there exists t ≥ 1 such that a /∈ ptA.(ii) dim VA = n− 1.

Lemma 5.5. µ(R1) = 1.

Proof. Let Γ1 ⊆ Γ be the Borel subset consisting of those ψ such that for eachz ∈ Zn(p), there exists t ≥ 1 such that ψ([z/pt]) 6= 0. Then it is easily checked thatν(Γ1) = 1. We shall show that

A ∈ R | π(A) = kerψ for some ψ ∈ Γ1 ⊆ R1.

So let ψ ∈ Γ1 and let π(A) = kerψ. Then it is clear that A satisfies condition 5.4(i).Let r ≥ 0 be the greatest integer such that a = en/p

r ∈ A. By Exercise 93.1 [12],in order to prove that dim VA = n − 1, it is enough to show that a is a p-basis ofA. Suppose not. Then there exists an element b = z/ps ∈ A, where z ∈ Zn(p) ands ≥ 0, such that (a, b) is a p-independent sequence. Since π(A) = kerψ, we musthave that:

(1) ψ([en/pr]) = ψ([z/ps]) = 0;(2) there exist integers 0 < m1,m2 < p such that

ψ([en/pr+1]) = [m1/p] and ψ([z/ps+1]) = [m2/p].

Let 0 < ` < p satisfy `m1 +m2 ≡ 0 (mod p). Then

ψ(`[en/pr+1] + [z/ps+1]) = 0

and so (`a+b)/p ∈ A, which contradicts the hypothesis that (a, b) is a p-independentsequence.

Let ψ ∈ Γ1 and let A ∈ R1 satisfy π(A) = kerψ. We shall next discuss therelationship between ψ = (β1, . . . , βn) ∈ Znp and the Qp-space VA. For each integer1 ≤ i ≤ n, let

βi = b(i)1 + b

(i)2 p+ · · ·+ b

(i)t pt−1 + · · ·

and let ri ≥ 0 be the greatest integer such that xi = ei/pri ∈ A. Then xn is a

p-basis of A; and for each 1 ≤ i ≤ n, we have that

αi = βi/pri = a

(i)1 + a

(i)2 p+ · · ·+ a

(i)t pt−1 + · · · ∈ Z∗p

is a p-adic unit and

ψ([xi/pt]) =

[a

(i)1 + a

(i)2 p+ · · ·+ a

(i)t pt−1

pt

].

For each 1 ≤ i ≤ n− 1, let

γi = −αi/αn = c(i)1 + c

(i)2 p+ · · ·+ c

(i)t pt−1 + · · · ∈ Z∗p.

Thent∑

r=1

a(i)r pr−1 + (

t∑r=1

c(i)r pr−1)(t∑

r=1

a(n)r pr−1) ≡ 0 (mod pt)

for each 1 ≤ i ≤ n− 1; and so

ψ([xi/pt] + (t∑

r=1

c(i)r pr−1)[xn/pt]) = 0.



Thus xi + (∑t

r=1 c(i)r pr−1)xn ∈ ptA for each t ≥ 1. Hence, arguing as in Section 4,

we see that(x1 + γ1xn, . . . , xn−1 + γn−1xn)

is a basis of VA.Now we shall begin our study of the structure of Aut(A) for A ∈ R1. First note

that the natural action of GLn(Q) on Qn extends canonically to an action on Qnp .Furthermore, if π ∈ Aut(A) 6 GLn(Q), then π(A) = A and so π(VA) = VA. Itwill also be useful to consider the corresponding contragredient representation ofGLn(Q) on the dual space D = Hom(Qnp ,Qp), defined by

(π.f)(v) = f(π−1v)

for π ∈ GLn(Q), f ∈ D and v ∈ Qnp . Recall that the linear transformation f 7→ π.f

is represented by the matrix (π−1)t relative to the dual basis e1, . . . , en. (Forexample, see [8, Section 43].) For subspaces U 6 Qnp and W 6 D, let

U⊥ = f ∈ D | f(u) = 0 for all u ∈ Uand

W⊥ = u ∈ Qnp | f(u) = 0 for all f ∈W.Notice that if π ∈ GLn(Q) satisfies π(U) = U , then π(U⊥) = U⊥.

Let ψ ∈ Γ1 and let A ∈ R1 satisfy π(A) = kerψ. Suppose that π ∈ Aut(A) 6GLn(Q). Then π(VA) = VA and so π(V ⊥A ) = V ⊥A . Since dimVA = n − 1, wehave that dimV ⊥A = 1. Let 0 6= f ∈ V ⊥A . Then π.f = λf for some eigenvalue0 6= λ ∈ Q ∩Qp of the matrix (π−1)t. Let E 6 D be the eigenspace correspondingto λ. We shall show that if A ∈ R1 is “sufficiently random”, then E = D andso (π−1)t = λI. Hence π = λI; and since A is not p-divisible, we must have thatλ ∈ Z∗(p).

So suppose that dimE = e < n. Since λ ∈ Q ∩Qp, it follows that E has a basisconsisting of functions f1, . . . , fe such that for each 1 ≤ k ≤ e,

fk = r(k)1 e1 + · · ·+ r(k)

n en

for some r(k)i ∈ Q∩Qp. This implies that E⊥∩ (Q∩Qp)n is an (n− e)-dimensional

vector space over Q ∩ Qp. In particular, since E⊥ 6 (V ⊥A )⊥ = VA, it follows thatthere exists a nonzero vector 0 6= v ∈ VA ∩ Q

n. Once again, for each 1 ≤ i ≤ n,

let xi = ei/pri, where ri ≥ 0 is the greatest integer such that ei ∈ priA. Then

v = q1x1 + · · ·+ qnxn for some qi ∈ Q ∩Qp. Since (x1 + γ1xn, . . . , xn−1 + γn−1xn)is a basis of VA, there exist θj ∈ Qp for 1 ≤ j ≤ n− 1 such that

q1x1 + · · ·+ qnxn = θ1(x1 + γ1xn) + · · ·+ θn−1(xn−1 + γn−1xn).

It follows that θj = qj for 1 ≤ j ≤ n− 1 and hence

qn = q1γ1 + · · ·+ qn−1γn−1.

Since γj = −αj/αn for 1 ≤ j ≤ n− 1, we obtain that

q1α1 + · · ·+ qnαn = 0.

Finally since αi = βi/pri for 1 ≤ i ≤ n, we see that β1, . . . , βn are linearly indepen-

dent over Q ∩ Qp. Hence in order to complete the proof of Theorem 5.1, we needonly show that if

Y = (β1, . . . , βn) ∈ Znp | β1, . . . , βn are linearly dependent over Q ∩Qp,


252 SIMON THOMAS

then ν(Y ) = 0. To see this, note that for each q = (q1, . . . , qn) ∈ (Q ∩ Zp)n, theclosed subgroup

Hq = (β1, . . . , βn) ∈ Znp | q1β1 + · · ·+ qnβn = 0

has infinite index in Znp and so ν(Hq) = 0. Hence

ν(Y ) = ν(⋃Hq | q ∈ (Q ∩ Zp)n) = 0.

In the remainder of this section, we shall restrict our attention to the class ofp-local groups of rank 2; and we shall point out some connections between this mate-rial and the main open problems on treeable countable Borel equivalence relations.We shall begin by presenting an alternative realisation (up to Borel bireducibility)of the isomorphism relation ∼=(p)

2 on S(p)(Q2).

Definition 5.6. For each p ∈ P, let E(p) be the orbit equivalence relation in-duced by the Borel action of GL2(Q) on Qp ∪ ∞ as a group of fractional lineartransformations; i.e. (

a bc d

).γ =

aγ + b

cγ + d

for each γ ∈ Qp ∪ ∞.

Theorem 5.7. (∼=(p)2 ) ∼B E(p) for each p ∈ P.

Proof. Let B(p)(Q2) be the Borel set consisting of those A ∈ S(p)(Q2) which satisfythe following conditions:

(i) A ∈ R(p)(Q2).(ii) dim VA = 1.

Claim 5.8. S(p)(Q2)rB(p)(Q2) is countable.

Proof of Claim 5.8. It is easily checked that S(p)(Q2)rR(p)(Q2) is countable; andTheorem 4.7 implies that there are only countably many A ∈ R(p)(Q2) such thatdimVA 6= 1.

It follows easily that (∼=(p)2 ) ∼B (∼=(p)

2 B(p)(Q2)). Next we shall show that(∼=(p)

2 B(p)(Q2)) ≤B E(p). To see this, let e1, e2 be the standard basis of the vectorspaceQ2

p. Then for each A ∈ B(p)(Q2), there exists a unique element γA ∈ Qp∪∞such that

VA = 〈 γAe1 + e2 〉;and an easy calculation shows that if π ∈ GL2(Q), then

Vπ(A) = π(VA) = 〈 (π.γA)e1 + e2 〉.

(Of course, 〈 ∞e1 + e2 〉 should be interpreted as 〈 e1 〉.) Let f : B(p)(Q2) →Qp ∪ ∞ be the Borel map defined by f(A) = γA. Then we have just seen thatif A ∼=(p)

2 B, then f(A)E(p)f(B). Conversely, suppose that f(A)E(p)f(B); say,π.γA = γB. Then Vπ(A) = VB and so by Theorem 4.7, π(A) ≈ B. By [12, Exercise32.5], each p-basis of A can be lifted to a p-basis of A/pA. It follows that |A/pA| = p

and so [12, Proposition 92.1] implies that A ∼=(p)2 B.



Finally we shall show that E(p) ≤B (∼=(p)2 B(p)(Q2)). For each γ ∈ Qp ∪ ∞,

let vγ = γe1 + e2 and let wγ = pre2, where prγ ∈ Z∗p. (For γ ∈ 0,∞, we setw0 = e1 and w∞ = e2.) Let g : Qp ∪∞ → B(p)(Q2) be the Borel map defined by

g(γ) = [Qpvγ ⊕ Zpwγ ] ∩Q2.

Then it is easily checked that Vg(γ) = Qpvγ . It follows that if γ1, γ2 ∈ Qp ∪ ∞,then γ1E

(p)γ2 iff g(γ1) ∼=(p)2 g(γ2).

Now let F2 be the free group on two generators. In [19], Kechris asked whetherfor every countable Borel equivalence relation E, the following statements are equiv-alent:

(a) E is non-hyperfinite.(b) There is a free action of F2 on a standard Borel space X , which admits an

invariant probability measure, such that EXF2≤B E.

Note that the relation EXF2in clause (b) is necessarily treeable and non-hyperfinite.

Thus a negative answer to the following question would also give a counterexampleto the preceeding question.

Question 5.9. Does there exist a non-hyperfinite treeable countable Borel equiv-alence relation E such that E ≤B (∼=(p)

2 )?

Another important open question asks whether there exists a (necessarily tree-able) countable Borel equivalence relation E such that E0 <B E <B ET∞. Ofcourse, this question would receive a positive answer if we could find two tree-able countable Borel equivalences which were incomparable with respect to Borelreducibility. We shall conclude this section by presenting some possible candidates.

Definition 5.10. For each p ∈ P, let E(p)T be the orbit equivalence relation in-

duced by the Borel action of GL2(Z) on Qp ∪ ∞ as a group of fractional lineartransformations.

Theorem 5.11. For each p ∈ P, the countable Borel equivalence relation E(p)T is

treeable and non-hyperfinite.

Proof. Since the subgroup ±I of GL2(Z) fixes each element of Qp ∪∞, we canalso regard E(p)

T as the orbit equivalence relation of the induced action of PGL2(Z)on Qp ∪ ∞. Suppose that γ ∈ Qp ∪ ∞ is fixed by some nonidentity elementϕ ∈ PGL2(Z). Then an easy calculation shows that γ ∈ Q ∪ ∞. In particular,PGL2(Z) acts freely on the complement of a countable subset of Qp ∪ ∞. ByJackson-Kechris-Louveau [16, Proposition 3.4], E(p)

T is treeable.By Jackson-Kechris-Louveau [16, Proposition 1.7], in order to show that E(p)

T isnon-hyperfinite, it is enough to show that there exists a PGL2(Z)-invariant proba-bility measure on Qp∪∞. As in the proof of Theorem 5.1, let R = R(p)(Q2,Z2

(p))be the standard Polish GLn(Z(p))-space consisting of those A ∈ R(Q2) such thatZn(p) 6 A; and let R1 ⊂ R be the Borel subset consisting of those A ∈ R whichsatisfy the following conditions:

(i) For all a ∈ A, there exists t ≥ 1 such that a /∈ ptA.(ii) dim VA = 1.


254 SIMON THOMAS

By Lemma 5.5, there exists a GLn(Z(p))-invariant probability measure µ on R1.As in the proof of Theorem 5.7, let f : R1 → Qp ∪ ∞ be the Borel map definedby f(A) = γA, where γA ∈ Qp ∪ ∞ is the unique element such that

VA = 〈 γAe1 + e2 〉.

Let µ = fµ be the probability measure defined on Qp ∪ ∞ by

µ(B) = µ(f−1(B))

for each Borel subset B ⊆ Qp ∪ ∞. Since f(π(A)) = π.f(A) for all A ∈ R1 andπ ∈ GL2(Z), it follows that µ is PGL2(Z)-invariant.

Conjecture 5.12. If p 6= q are distinct primes, then E(p)T B E

(q)T .

Of course, it is also possible to define a Borel action of GL2(Z) on R ∪ ∞as a group of fractional linear transformations. However, Jackson-Kechris-Louveau[16] have pointed out that, in this case, the induced orbit equivalence relation ishyperfinite.

6. A cocycle reduction result

In this section, we shall prove Theorem 2.3. As we mentioned earlier, this resultis a straightforward consequence of Zimmer’s superrigidity theorem [34, Theorem5.2.5] and the ideas of Adams-Kechris [3]. First we need to recall some notionsfrom valuation theory. (A clear account of this material can be found in Margulis[26, Chapter I].) Let F be an algebraic number field; i.e. a finite extension of thefield Q of rational numbers. Let R be the set of all non-equivalent valuations ofF and let R∞ ⊂ R be the set of archimedean valuations. For each ν ∈ R, let Fνbe the completion of F relative to ν. If ν ∈ R∞, then Fν = R or Fν = C; and ifν ∈ RrR∞, then Fν is a totally disconnected local field; i.e. a finite extension ofthe field Qp of p-adic numbers for some prime p. In particular, each Fν is a localfield.

Let S ⊆ R be a set of valuations of F . Then an element x ∈ F is said to beS-integral iff |x|ν ≤ 1 for each non-archimedean valuation ν /∈ S. The set of allS-integral elements is a subring of F , which will be denoted by F (S). Furthermore,F is the union of the subrings F (S), where S ranges over the finite sets of valuationsof the field F .

The proof of Theorem 2.3 will be based upon the following cocycle reductiontheorem.

Theorem 6.1. Let m ≥ 3 and let Y be a standard Borel SLm(R)-space with aninvariant ergodic probability measure. Let K be a local field and let T be a simplealgebraic K-group such that dimT < m2 − 1. Then for every Borel cocycle α :SLm(R)×Y → T (K), there exists an equivalent cocycle γ such that γ(SLm(R)×Y )is contained in a compact subgroup of T (K).

Proof. If we add the hypothesis that α is not equivalent to a cocycle β taking valuesin a subgroup of the form L(K) for some proper algebraic K-subgroup L of T , thenTheorem 6.1 is an immediate consequence of Zimmer [34, Theorem 5.2.5]. Butarguing as in the proof of Adams-Kechris [3, Theorem 3.5], we can easily reduceour analysis to the case when this extra hypothesis holds.



In the proof of Theorem 2.3, we shall also make use of the notions of an inducedaction and an induced cocycle, which are defined as follows. (A fuller account ofthese notions can be found in Adams-Kechris [3, Section 2].) Suppose that G is alcsc group and that Γ is a closed subgroup of G. Let G/Γ be the set of left cosetsof Γ in G. By the Effros-Mackey cross section theorem [28, Theorem 5.4.2], thereexists a Borel transversal T for G/Γ; i.e. a Borel subset T ⊆ G which meets everycoset in a unique element. Fix such a Borel transversal with 1 ∈ T . Then we canidentify T with G/Γ by identifying t with tΓ; and then the action of G on G/Γinduces a corresponding Borel action of G on T , defined by

g.t = the unique element in T ∩ gtΓ.

The associated strict cocycle ρ : G× T → Γ is defined by

ρ(g, t) = the unique ϕ ∈ Γ such that (g.t)ϕ = gt

= (g.t)−1gt.

Now suppose that X is a standard Borel Γ-space with an invariant ergodic proba-bility measure µ. Suppose also that Γ is a lattice in G; i.e. Γ is a discrete subgroupof G and the action of G on G/Γ admits an invariant probability measure. Let νbe the corresponding invariant probability measure on T . Then the induced actionof G on the standard Borel space Y = X × T is defined by

g.(x, t) = (ρ(g, t).x, g.t);

and it is easily checked that µ×ν is an invariant ergodic probability measure on Y .(See Adams-Kechris [3, Section 2].) Finally given any strict cocycle β : Γ×X → H ,the corresponding induced cocycle β : G× Y → H is defined by

β(g, (x, t)) = β(ρ(g, t), x).

In the proof of Theorem 2.3, these notions will be used in the case when Γ = SLm(Z)and G = SLm(R). We shall suppress explicit mention of the Borel transversal Tand instead write

Y = X × (SLm(R)/SLm(Z)).

We are now ready to begin the proof of Theorem 2.3. So let m ≥ 3 and let Xbe a standard Borel SLm(Z)-space with an invariant ergodic probability measureµ. Suppose that G is an algebraic Q-group such that dimG < m2 − 1 and thatH 6 G(Q). Let α : SLm(Z) × X → H be a Borel cocycle. Then we can view αas a cocycle taking values in G(Q). Furthermore, after passing to a finite ergodicextension of X if necessary, we can suppose that G is connected. (For example,see [3, Propositions 2.5 and 2.6].) Let R be the soluble radical of G. Then G/Ris a connected semisimple algebraic Q-group; and, in particular, G/R has a finitecentre A/R. Let G = G/A. Then there exist simple algebraic Q-groups T1, . . . , Tksuch that G = T1 × · · · × Tk. Now consider the Borel cocycle

α : SLm(Z)×X → G(Q) = T1(Q)× · · · × Tk(Q)

defined by α = π α, where π : G(Q) → G(Q) be the canonical surjection. SinceSLm(Z) is a Kazhdan group, [35, Lemma 2.2] implies that α is equivalent to acocycle β taking values in a finitely generated subgroup Λ of G(Q). So there exists


256 SIMON THOMAS

an algebraic number field F and a finite set S of valuations of F such that:

(i) each Ti is an algebraic F -group; and(ii) Λ 6 G(F (S)) = T1(F (S))× · · · × Tk(F (S)).

Clearly we can suppose that S contains the set R∞ of archimedean valuations of F .It follows that if G(F (S)) is identified with its image under the diagonal embeddinginto

GS =∏ν∈S

G(Fν) =k∏i=1

∏ν∈S

Ti(Fν),

then G(F (S)) is a discrete subgroup of GS . (For example, see [26, Section I.3.2].)By Zimmer [34, Theorem B.9], we can suppose that β : SLm(Z) × X → Λ isa strict cocycle. Consider the induced Borel action of SLm(R) on Y = X ×(SLm(R)/SLm(Z)) and let β : SLm(R) × Y → Λ be the corresponding inducedcocycle. For each ν ∈ S and 1 ≤ i ≤ k, let pνi : GS → Ti(Fν) be the canonicalprojection; and, viewing β as a cocycle into GS , let βνi : SLm(R) × Y → Ti(Fν)be the Borel cocycle defined by βνi = pνi β. By Theorem 6.1, βνi is equivalent toa cocycle taking values in a compact subgroup Cνi of Ti(Fν). It follows that β isequivalent to a cocycle taking values in the compact subgroup C =

∏ki=1

∏ν∈S Cνi

of GS . By Adams-Kechris [3, Proposition 2.4], there exists g ∈ GS and a cocycleγ : SLm(R) × Y → Λ such that γ ∼ β and γ takes values in the finite subgroupB = Λ∩gCg−1 of Λ; and hence by Adams-Kechris [3, Proposition 2.3], there existsa cocycle γ : SLm(Z) ×X → Λ such that γ ∼ β and γ also takes values in B. LetB = π−1(B). Then B is a soluble-by-finite subgroup of G(Q); and since α ∼ γ,it follows that α is equivalent to a cocycle γ : SLm(Z) ×X → G(Q) taking valuesin B. Since SLm(Z) is a Kazhdan group and B is a countable amenable group,Zimmer [34, Theorem 9.1.1] implies that γ is equivalent to a cocycle taking valuesin a finite subgroup of B. Applying Adams-Kechris [3, Proposition 2.4] once more,it follows that α is equivalent to a cocycle γ : SLm(Z)×X → H taking values in afinite subgroup of H . This completes the proof of Theorem 2.3.

Acknowledgements

I would like to thank David Arnold, Greg Hjorth, Alexander Kechris and BobanVelickovic for very helpful discussions concerning the material in this paper.

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73 (1991), 151–159. MR 93e:22012

Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscat-

away, New Jersey 08854-8019

E-mail address: [email protected]



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